# American Institute of Mathematical Sciences

December  2019, 9(4): 793-836. doi: 10.3934/mcrf.2019050

## Local null controllability of a rigid body moving into a Boussinesq flow

 1 TIFR Centre for Applicable Mathematics, Post Bag No. 6503, GKVK Post Office, Bangalore 560065, India 2 Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France

* Corresponding author: Takéo Takahashi

Received  January 2018 Revised  September 2019 Published  November 2019

In this paper, we study the controllability of a fluid-structure interaction system. We consider a viscous and incompressible fluid modeled by the Boussinesq system and the structure is a rigid body with arbitrary shape which satisfies Newton's laws of motion. We assume that the motion of this system is bidimensional in space. We prove the local null controllability for the velocity and temperature of the fluid and for the position and velocity of rigid body for a control acting only on the temperature equation on a fixed subset of the fluid domain.

Citation: Arnab Roy, Takéo Takahashi. Local null controllability of a rigid body moving into a Boussinesq flow. Mathematical Control and Related Fields, 2019, 9 (4) : 793-836. doi: 10.3934/mcrf.2019050
##### References:
 [1] M. Badra and T. Takahashi, Feedback stabilization of a fluid-rigid body interaction system, Adv. Differential Equations, 19 (2014), 1137–1184, URL http://projecteuclid.org/euclid.ade/1408367290. [2] A. Bensoussan, G. Da Prato, M. C. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, Springer Science & Business Media, 2007. doi: 10.1007/978-0-8176-4581-6. [3] M. Boulakia and S. Guerrero, Local null controllability of a fluid-solid interaction problem in dimension 3, J. Eur. Math. Soc. (JEMS), 15 (2013), 825-856.  doi: 10.4171/JEMS/378. [4] M. Boulakia and A. Osses, Local null controllability of a two-dimensional fluid-structure interaction problem, ESAIM Control Optim. Calc. Var., 14 (2008), 1-42.  doi: 10.1051/cocv:2007031. [5] N. Carreño, Local controllability of the $N$-dimensional Boussinesq system with $N-1$ scalar controls in an arbitrary control domain, Math. Control Relat. Fields, 2 (2012), 361-382.  doi: 10.3934/mcrf.2012.2.361. [6] N. Carreño and S. Guerrero, Local null controllability of the $N$-dimensional Navier-Stokes system with $N-1$ scalar controls in an arbitrary control domain, J. Math. Fluid Mech., 15 (2013), 139-153.  doi: 10.1007/s00021-012-0093-2. [7] C. Conca, H. J. San Martin and M. Tucsnak, Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations, 25 (2000), 1019-1042. [8] J.-M. Coron, On the controllability of the $2$-D incompressible Navier-Stokes equations with the Navier slip boundary conditions, ESAIM Contrôle Optim. Calc. Var., 1 (1995/96), 35-75.  doi: 10.1051/cocv:1996102. [9] J.-M. Coron and S. Guerrero, Null controllability of the $N$-dimensional Stokes system with $N-1$ scalar controls, J. Differential Equations, 246 (2009), 2908-2921.  doi: 10.1016/j.jde.2008.10.019. [10] J.-M. Coron and P. Lissy, Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components, Invent. Math., 198 (2014), 833-880.  doi: 10.1007/s00222-014-0512-5. [11] B. Desjardins and M. J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146 (1999), 59-71.  doi: 10.1007/s002050050136. [12] A. Doubova and E. Fernández-Cara, Some control results for simplified one-dimensional models of fluid-solid interaction, Math. Models Methods Appl. Sci., 15 (2005), 783-824.  doi: 10.1142/S0218202505000522. [13] C. Fabre and G. Lebeau, Prolongement unique des solutions de l'equation de Stokes, Comm. Partial Differential Equations, 21 (1996), 573-596.  doi: 10.1080/03605309608821198. [14] E. Feireisl, On the motion of rigid bodies in a viscous fluid, Appl. Math., 47 (2002), 463–484, Mathematical theory in fluid mechanics (Paseky, 2001). doi: 10.1023/A:1023245704966. [15] E. Feireisl, On the motion of rigid bodies in a viscous compressible fluid, Arch. Ration. Mech. Anal., 167 (2003), 281-308.  doi: 10.1007/s00205-002-0242-5. [16] E. Fernández-Cara, M. González-Burgos, S. Guerrero and J.-P. Puel, Null controllability of the heat equation with boundary Fourier conditions: the linear case, ESAIM Control Optim. Calc. Var., 12 (2006), 442-465.  doi: 10.1051/cocv:2006010. [17] E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1399-1446.  doi: 10.1137/S0363012904439696. [18] E. Fernández-Cara, S. Guerrero, O. Y. Imanuvilov and J.-P. Puel, Local exact controllabilityof the Navier-Stokes system, J. Math. Pures Appl. (9), 83 (2004), 1501–1542. doi: 10.1016/j.matpur.2004.02.010. [19] E. Fernández-Cara, S. Guerrero, O. Y. Imanuvilov and J.-P. Puel, Some controllability results for the $N$-dimensional Navier-Stokes and Boussinesq systems with $N-1$ scalar controls, SIAM J. Control Optim., 45 (2006), 146-173.  doi: 10.1137/04061965X. [20] A. V. Fursikov and O. Y. Emanuilov, Èxact controllability of the Navier-Stokes and Boussinesq equations, Uspekhi Mat. Nauk, 54 (1999), 93–146; translation in Russian Math. Surveys, 54 (1999), 565–618. doi: 10.1070/rm1999v054n03ABEH000153. [21] A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, vol. 34 of Lecture Notes Series, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. [22] A. V. Fursikov and O. Y. Imanuvilov, Local exact boundary controllability of the Boussinesq equation, SIAM J. Control Optim., 36 (1998), 391-421.  doi: 10.1137/S0363012996296796. [23] G. P. Galdi, On the steady self-propelled motion of a body in a viscous incompressible fluid, Arch. Ration. Mech. Anal., 148 (1999), 53-88.  doi: 10.1007/s002050050156. [24] G. P. Galdi and A. L. Silvestre, Strong solutions to the problem of motion of a rigid body in a Navier-Stokes liquid under the action of prescribed forces and torques, in Nonlinear Problems in Mathematical Physics and Related Topics, I, vol. 1 of Int. Math. Ser. (N. Y.), Kluwer/Plenum, New York, 2002, 121–144. doi: 10.1007/978-1-4615-0777-2_8. [25] M. González-Burgos, S. Guerrero and J.-P. Puel, Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation, Commun. Pure Appl. Anal., 8 (2009), 311-333.  doi: 10.3934/cpaa.2009.8.311. [26] C. Grandmont and Y. Maday, Existence for an unsteady fluid-structure interaction problem, M2AN Math. Model. Numer. Anal., 34 (2000), 609-636.  doi: 10.1051/m2an:2000159. [27] S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 29–61, . doi: 10.1016/j.anihpc.2005.01.002. [28] M. D. Gunzburger, H.-C. Lee and G. A. Seregin, Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions, J. Math. Fluid Mech., 2 (2000), 219-266.  doi: 10.1007/PL00000954. [29] O. Imanuvilov and T. Takahashi, Exact controllability of a fluid-rigid body system, J. Math. Pures Appl. (9), 87 (2007), 408–437. doi: 10.1016/j.matpur.2007.01.005. [30] O. Y. Imanuvilov, Local exact controllability for the $2$-D Navier-Stokes equations with the Navier slip boundary conditions, in Turbulence Modeling and Vortex Dynamics (Istanbul, 1996), vol. 491 of Lecture Notes in Phys., Springer, Berlin, 1997, 148–168. doi: 10.1007/BFb0105035. [31] O. Y. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), 39-72.  doi: 10.1051/cocv:2001103. [32] O. Y. Imanuvilov, J. P. Puel and M. Yamamoto, Carleman estimates for parabolic equations with nonhomogeneous boundary conditions, Chin. Ann. Math. Ser. B, 30 (2009), 333-378.  doi: 10.1007/s11401-008-0280-x. [33] J.-L. Lions and E. Zuazua, A generic uniqueness result for the Stokes system and its control theoretical consequences, in Partial Differential Equations and Applications, vol. 177 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1996, 221–235. [34] Y. Liu, T. Takahashi and M. Tucsnak, Single input controllability of a simplified fluid-structure interaction model, ESAIM Control Optim. Calc. Var., 19 (2013), 20-42.  doi: 10.1051/cocv/2011196. [35] S. Nečasová, On the motion of several rigid bodies in an incompressible non-Newtonian and heat-conducting fluid, Ann. Univ. Ferrara Sez. VII Sci. Mat., 55 (2009), 325-352.  doi: 10.1007/s11565-009-0085-1. [36] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [37] J.-P. Raymond and M. Vanninathan, Exact controllability in fluid-solid structure: The Helmholtz model, ESAIM Control Optim. Calc. Var., 11 (2005), 180-203.  doi: 10.1051/cocv:2005006. [38] J. A. San Martin, V. Starovoitov and M. Tucsnak, Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal., 161 (2002), 113-147.  doi: 10.1007/s002050100172. [39] D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence, Japan J. Appl. Math., 4 (1987), 99-110.  doi: 10.1007/BF03167757. [40] T. Takahashi, Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain, Adv. Differential Equations, 8 (2003), 1499-1532. [41] T. Takahashi and M. Tucsnak, Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid, J. Math. Fluid Mech., 6 (2004), 53-77.  doi: 10.1007/s00021-003-0083-4. [42] R. Temam, Navier-Stokes Equations, vol. 2 of Studies in Mathematics and its Applications, Revised edition, North-Holland Publishing Co., Amsterdam-New York, 1979, Theory and numerical analysis, With an appendix by F. Thomasset. [43] R. Temam, Problèmes Mathématiques en Plasticité, vol. 12 of Méthodes Mathématiques de l'Informatique [Mathematical Methods of Information Science], Gauthier-Villars, Montrouge, 1983. [44] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

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##### References:
 [1] M. Badra and T. Takahashi, Feedback stabilization of a fluid-rigid body interaction system, Adv. Differential Equations, 19 (2014), 1137–1184, URL http://projecteuclid.org/euclid.ade/1408367290. [2] A. Bensoussan, G. Da Prato, M. C. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, Springer Science & Business Media, 2007. doi: 10.1007/978-0-8176-4581-6. [3] M. Boulakia and S. Guerrero, Local null controllability of a fluid-solid interaction problem in dimension 3, J. Eur. Math. Soc. (JEMS), 15 (2013), 825-856.  doi: 10.4171/JEMS/378. [4] M. Boulakia and A. Osses, Local null controllability of a two-dimensional fluid-structure interaction problem, ESAIM Control Optim. Calc. Var., 14 (2008), 1-42.  doi: 10.1051/cocv:2007031. [5] N. Carreño, Local controllability of the $N$-dimensional Boussinesq system with $N-1$ scalar controls in an arbitrary control domain, Math. Control Relat. Fields, 2 (2012), 361-382.  doi: 10.3934/mcrf.2012.2.361. [6] N. Carreño and S. Guerrero, Local null controllability of the $N$-dimensional Navier-Stokes system with $N-1$ scalar controls in an arbitrary control domain, J. Math. Fluid Mech., 15 (2013), 139-153.  doi: 10.1007/s00021-012-0093-2. [7] C. Conca, H. J. San Martin and M. Tucsnak, Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations, 25 (2000), 1019-1042. [8] J.-M. Coron, On the controllability of the $2$-D incompressible Navier-Stokes equations with the Navier slip boundary conditions, ESAIM Contrôle Optim. Calc. Var., 1 (1995/96), 35-75.  doi: 10.1051/cocv:1996102. [9] J.-M. Coron and S. Guerrero, Null controllability of the $N$-dimensional Stokes system with $N-1$ scalar controls, J. Differential Equations, 246 (2009), 2908-2921.  doi: 10.1016/j.jde.2008.10.019. [10] J.-M. Coron and P. Lissy, Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components, Invent. Math., 198 (2014), 833-880.  doi: 10.1007/s00222-014-0512-5. [11] B. Desjardins and M. J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146 (1999), 59-71.  doi: 10.1007/s002050050136. [12] A. Doubova and E. Fernández-Cara, Some control results for simplified one-dimensional models of fluid-solid interaction, Math. Models Methods Appl. Sci., 15 (2005), 783-824.  doi: 10.1142/S0218202505000522. [13] C. Fabre and G. Lebeau, Prolongement unique des solutions de l'equation de Stokes, Comm. Partial Differential Equations, 21 (1996), 573-596.  doi: 10.1080/03605309608821198. [14] E. Feireisl, On the motion of rigid bodies in a viscous fluid, Appl. Math., 47 (2002), 463–484, Mathematical theory in fluid mechanics (Paseky, 2001). doi: 10.1023/A:1023245704966. [15] E. Feireisl, On the motion of rigid bodies in a viscous compressible fluid, Arch. Ration. Mech. Anal., 167 (2003), 281-308.  doi: 10.1007/s00205-002-0242-5. [16] E. Fernández-Cara, M. González-Burgos, S. Guerrero and J.-P. Puel, Null controllability of the heat equation with boundary Fourier conditions: the linear case, ESAIM Control Optim. Calc. Var., 12 (2006), 442-465.  doi: 10.1051/cocv:2006010. [17] E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1399-1446.  doi: 10.1137/S0363012904439696. [18] E. Fernández-Cara, S. Guerrero, O. Y. Imanuvilov and J.-P. Puel, Local exact controllabilityof the Navier-Stokes system, J. Math. Pures Appl. (9), 83 (2004), 1501–1542. doi: 10.1016/j.matpur.2004.02.010. [19] E. Fernández-Cara, S. Guerrero, O. Y. Imanuvilov and J.-P. Puel, Some controllability results for the $N$-dimensional Navier-Stokes and Boussinesq systems with $N-1$ scalar controls, SIAM J. Control Optim., 45 (2006), 146-173.  doi: 10.1137/04061965X. [20] A. V. Fursikov and O. Y. Emanuilov, Èxact controllability of the Navier-Stokes and Boussinesq equations, Uspekhi Mat. Nauk, 54 (1999), 93–146; translation in Russian Math. Surveys, 54 (1999), 565–618. doi: 10.1070/rm1999v054n03ABEH000153. [21] A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, vol. 34 of Lecture Notes Series, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. [22] A. V. Fursikov and O. Y. Imanuvilov, Local exact boundary controllability of the Boussinesq equation, SIAM J. Control Optim., 36 (1998), 391-421.  doi: 10.1137/S0363012996296796. [23] G. P. Galdi, On the steady self-propelled motion of a body in a viscous incompressible fluid, Arch. Ration. Mech. Anal., 148 (1999), 53-88.  doi: 10.1007/s002050050156. [24] G. P. Galdi and A. L. Silvestre, Strong solutions to the problem of motion of a rigid body in a Navier-Stokes liquid under the action of prescribed forces and torques, in Nonlinear Problems in Mathematical Physics and Related Topics, I, vol. 1 of Int. Math. Ser. (N. Y.), Kluwer/Plenum, New York, 2002, 121–144. doi: 10.1007/978-1-4615-0777-2_8. [25] M. González-Burgos, S. Guerrero and J.-P. Puel, Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation, Commun. Pure Appl. Anal., 8 (2009), 311-333.  doi: 10.3934/cpaa.2009.8.311. [26] C. Grandmont and Y. Maday, Existence for an unsteady fluid-structure interaction problem, M2AN Math. Model. Numer. Anal., 34 (2000), 609-636.  doi: 10.1051/m2an:2000159. [27] S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 29–61, . doi: 10.1016/j.anihpc.2005.01.002. [28] M. D. Gunzburger, H.-C. Lee and G. A. Seregin, Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions, J. Math. Fluid Mech., 2 (2000), 219-266.  doi: 10.1007/PL00000954. [29] O. Imanuvilov and T. Takahashi, Exact controllability of a fluid-rigid body system, J. Math. Pures Appl. (9), 87 (2007), 408–437. doi: 10.1016/j.matpur.2007.01.005. [30] O. Y. Imanuvilov, Local exact controllability for the $2$-D Navier-Stokes equations with the Navier slip boundary conditions, in Turbulence Modeling and Vortex Dynamics (Istanbul, 1996), vol. 491 of Lecture Notes in Phys., Springer, Berlin, 1997, 148–168. doi: 10.1007/BFb0105035. [31] O. Y. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), 39-72.  doi: 10.1051/cocv:2001103. [32] O. Y. Imanuvilov, J. P. Puel and M. Yamamoto, Carleman estimates for parabolic equations with nonhomogeneous boundary conditions, Chin. Ann. Math. Ser. B, 30 (2009), 333-378.  doi: 10.1007/s11401-008-0280-x. [33] J.-L. Lions and E. Zuazua, A generic uniqueness result for the Stokes system and its control theoretical consequences, in Partial Differential Equations and Applications, vol. 177 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1996, 221–235. [34] Y. Liu, T. Takahashi and M. Tucsnak, Single input controllability of a simplified fluid-structure interaction model, ESAIM Control Optim. Calc. Var., 19 (2013), 20-42.  doi: 10.1051/cocv/2011196. [35] S. Nečasová, On the motion of several rigid bodies in an incompressible non-Newtonian and heat-conducting fluid, Ann. Univ. Ferrara Sez. VII Sci. Mat., 55 (2009), 325-352.  doi: 10.1007/s11565-009-0085-1. [36] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [37] J.-P. Raymond and M. Vanninathan, Exact controllability in fluid-solid structure: The Helmholtz model, ESAIM Control Optim. Calc. Var., 11 (2005), 180-203.  doi: 10.1051/cocv:2005006. [38] J. A. San Martin, V. Starovoitov and M. Tucsnak, Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal., 161 (2002), 113-147.  doi: 10.1007/s002050100172. [39] D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence, Japan J. Appl. Math., 4 (1987), 99-110.  doi: 10.1007/BF03167757. [40] T. Takahashi, Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain, Adv. Differential Equations, 8 (2003), 1499-1532. [41] T. Takahashi and M. Tucsnak, Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid, J. Math. Fluid Mech., 6 (2004), 53-77.  doi: 10.1007/s00021-003-0083-4. [42] R. Temam, Navier-Stokes Equations, vol. 2 of Studies in Mathematics and its Applications, Revised edition, North-Holland Publishing Co., Amsterdam-New York, 1979, Theory and numerical analysis, With an appendix by F. Thomasset. [43] R. Temam, Problèmes Mathématiques en Plasticité, vol. 12 of Méthodes Mathématiques de l'Informatique [Mathematical Methods of Information Science], Gauthier-Villars, Montrouge, 1983. [44] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.
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